Category O

Remarks

Example: a Verma module $M(\lambda)$ lies in $\mathcal{O}$. It is generated by one vector, it is the direct sum of its weight spaces, and its weights lie in $\lambda - \Z_{\ge 0}\Phi^+$ with finite-dimensional weight spaces. Let $v$ be a weight vector of weight $\mu$. Then $U(\mathfrak{n}^+) v$ lies in the sum of the finitely many weight spaces of weights between $\mu$ and $\lambda$, so it is finite-dimensional. Every finite-dimensional $\mathfrak{g}$-module also lies in $\mathcal{O}$. Non-example: with $\mathfrak{b}^- = \mathfrak{h} \oplus \mathfrak{n}^-$, the lowest weight module $U(\mathfrak{g}) \otimes_{U(\mathfrak{b}^-)} \C_\lambda$ satisfies clauses 1 and 2. It fails clause 3, since $U(\mathfrak{n}^+)$ acts freely on its generator, so $U(\mathfrak{n}^+) v$ is infinite-dimensional. In $\mathcal{O}$ each weight space is finite-dimensional, each module has finite length, and the simple objects are the modules $L(\lambda)$, $\lambda \in \mathfrak{h}^*$. $\mathcal{O}$ is abelian, has enough projectives, and is the direct sum of subcategories indexed by central characters.